This chapter reviews selected topics in network systems and mathematical sociology. We first review classic results on Perron–Frobenius and algebraic graph theory. We then focus on mathematical sociology and describe models of opinion dynamics in social influence systems, including the classic French–Harary–DeGroot and the Friedkin–Johnsen models. Based on recent controlled experiments, we present recent empirical results on opinion dynamics along single issues and sequences of issues. Finally, motivated by these empirical results, we describe some mathematical models for the evolution of social power and influence systems via the reflected appraisal mechanism.